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11
Two extensions of Lubinsky’s universality theorem
 J. Anal. Math. 105
, 2008
"... Abstract. We extend some remarkable recent results of Lubinsky and Levin– Lubinsky from [−1, 1] to allow discrete eigenvalues outside σess and to allow σess first to be a finite union of closed intervals and then a fairly general compact set in R (one which is regular for the Dirichlet problem). 1 ..."
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Cited by 20 (9 self)
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Abstract. We extend some remarkable recent results of Lubinsky and Levin– Lubinsky from [−1, 1] to allow discrete eigenvalues outside σess and to allow σess first to be a finite union of closed intervals and then a fairly general compact set in R (one which is regular for the Dirichlet problem). 1
Equilibrium measures and capacities in spectral theory
, 2007
"... This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators wh ..."
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Cited by 15 (8 self)
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This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger operators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete–Szegő theory.
The impact of Stieltjes’ work on continued fractions and orthogonal polynomials
, 1993
"... Stieltjes’ work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes’ ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials. ..."
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Cited by 9 (0 self)
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Stieltjes’ work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes’ ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials.
Small Polynomials With Integer Coefficients
"... this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively wi ..."
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Cited by 8 (6 self)
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this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coe#cients. The problem of minimizing the uniform norm on E by monic polynomials from P n (C) is well known as the Chebyshev problem (see [4], [31], [43], [16], etc.) In the classical case E = [1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: T n (x) := 2 1n cos(n arccos x), n # N. Using a change of variable, we can immediately extend this to an arbitrary interval [a, b] # R, so that t n (x) := # b  a 2 # n T n # 2x  a  b b  a # is a monic polynomial with real coe#cients and the smallest uniform norm on [a, b] among all monic polynomials from P n (C). In fact, (1.1) #t n # [a,b] = 2 # b  a 4 # n , n # N, and we find that the Chebyshev constant for [a, b] is given by (1.2) t C ([a, b]) := lim n## #t n # 1/n [a,b] = b  a 4 . The Chebyshev constant of an arbitrary compact set E # C is defined in a similar fashion: (1.3) t C (E) := lim n## #t n # 1/n E , where t n is the Chebyshev polynomial of degree n on E. It is known that t C (E) is equal to the transfinite diameter and the logarithmic capacity cap(E) of the set E (cf. [43, pp. 7175], [16] and [30] for the definitions and background material). 2000 Mathematics Subject Classification. Primary 11C08, 30C10; Secondary 31A05, 31A15. Key words and phrases. Chebyshev polynomials, integer Chebyshev constant, integer transfinite diameter, zeros, multiple factors, asymptotic...
Green’s functions for multiply connected domains via conformal mapping
 SIAM Review
, 1999
"... Abstract. A method is described for the computation of the Green’s function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz–Christoffel conformal map of the pa ..."
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Cited by 8 (2 self)
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Abstract. A method is described for the computation of the Green’s function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz–Christoffel conformal map of the part of the upper halfplane exterior to the problem domain onto a semiinfinite strip whose end contains K − 1 slits. From the Green’s function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iterations. By making the end of the strip jagged, the method can be generalized to weighted Green’s functions and weighted approximations. Key words. Green’s function, conformal mapping, Schwarz–Christoffel formula, polynomial approximation,
The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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Cited by 4 (4 self)
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.
EQUIDISTRIBUTION OF POINTS VIA ENERGY
"... Abstract. We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates ..."
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Cited by 3 (3 self)
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Abstract. We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin’s constant, and problems on the means of zeros of polynomials with integer coefficients. 1. Asymptotic equidistribution of discrete sets Let E be a compact set in the complex plane C. Given a set of points Zn = {zk,n} n k=1 ⊂ C, n ≥ 2, the associated Vandermonde determinant is
THE MULTIVARIATE INTEGER CHEBYSHEV PROBLEM
"... Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivar ..."
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Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the HilbertFekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single variable polynomials in the complex plane, our estimate coincides with the HilbertFekete result. 1. The integer Chebyshev problem and its multivariate counterpart The supremum norm on a compact set E ⊂ C d, d ∈ N, is defined by �f�E: = sup f(z). z∈E We study polynomials with integer coefficients that minimize the sup norm on a set E, and investigate their asymptotic behavior. The univariate case (d = 1) has a long history, but the problem is virtually untouched for d ≥ 2. Let Pn(C) and Pn(Z) be the classes of algebraic polynomials in one variable, of degree at most n, respectively with complex and with integer coefficients. The problem of minimizing the uniform norm on E by monic polynomials from Pn(C) is the classical Chebyshev problem (see [5], [23], [26], etc.) For E = [−1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: Tn(x): = 2 1−n cos(n arccos x), n ∈ N. By a linear change of variable, we immediately obtain that � �n � � b − a 2x − a − b tn(x):=
DISTRIBUTION OF POINT CHARGES WITH SMALL DISCRETE ENERGY
"... Abstract. We study the asymptotic equidistribution of points near arbitrary compact sets of positive capacity in R d, d ≥ 2. Our main tools are the energy estimates for Riesz potentials. We also consider the quantitative aspects of this equidistribution in the classical Newtonian case. In particular ..."
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Abstract. We study the asymptotic equidistribution of points near arbitrary compact sets of positive capacity in R d, d ≥ 2. Our main tools are the energy estimates for Riesz potentials. We also consider the quantitative aspects of this equidistribution in the classical Newtonian case. In particular, we quantify the weak convergence of discrete measures to the equilibrium measure, and give the estimates of convergence rates for discrete potentials to the equilibrium potential. 1. Asymptotic equidistribution of discrete sets Let E be a compact set in R d, d ≥ 2. Denote the Euclidean distance between x ∈ R d and y ∈ R d by x − y. We consider potential theory associated with Riesz kernels kα(x): = x  α−d, x ∈ R d, 0 < α < d. For a Borel measure µ with compact support, define its energy by Iα[µ]: = kα(x − y) dµ(x)dµ(y). A central theme in potential theory is the study of the minimum energy problem Wα(E):= inf µ∈M(E) Iα[µ], where M(E) is the space of all positive unit Borel measures supported on E. If Robin’s constant Wα(E) is finite, then the above infimum is attained by the equilibrium measure µE ∈ M(E) [14, p. 131–133], which is a unique probability measure expressing the steady state distribution of charge on the conductor E. The capacity of E is defined by